Dimensional Analysis - Schrödinger equation

[astro_girl]

To illustrate the abstract reduction to dimensionless quantities apply it to the harmonic oscillator
V(x) = (m \omega^2 x^2) / 2
using x_0 = sqrt(h-bar/(m \omega))
and fi nd a dimensionless Schrodinger equation. Translate the known solutions to the Schrodinger
equation for the harmonic oscillator E_n = (n + 1/2)h-bar\omega
into the allowed energies ~E of the dimensionless Schrodinger equation.

I know this has to do with dimensional analysis, but I was sick when we had that class, and I've been searching for help on the internet the whole day without any luck. I don't think it is too difficult, I just don't really get what I have to do.

I guess the \omega and the m have to go, but how?

 

[mfb]

You can use LaTeX if you surround your equations by double $ or double # (for inline mode).

Can you re-write V in such a way that it just has factors of (x/x_0) instead of x?
Can you do something similar for the energies?

 

[Simon Bridge]

Welcome to PF;
I agree with mfb, but would put it differently...

I guess the \omega and the m have to go, but how?

You eliminate them using your simultaneous equations.
LaTeX makes the equations clearer like this:

(1) $$V(x) = \frac{1}{2} m \omega^2 x^2$$(2) $$x_0=\sqrt{ \frac{\hbar}{m\omega} }$$(are those correct?) i.e. for (2) don't you want ##x_0:V(x_0)=\frac{1}{2}\hbar## ?

Initially it looks like you have two equations and three unknowns - but ... notice how ##m\omega## always appear together in these equations? Maybe you can eliminate them together? (well... sort of)

... anyway - there's a third equation:
(3) $$E_n=\cdots$$

It will lead you down mfb's line... you are looking for ##V/E_n## in terms of ##x/x_0## ... treated that way you have three equations with three unknowns ... you just have to identify the unknowns. This you do by experience and you get that by trial and error.To see how I typeset the equations (essential for QM and useful elsewhere) just click on the "quote" button at the bottom of this post ;)